这个问题整了整整两天，终于解决了。记录下来。
1.文件下载
首先下载cors压缩包，解压，得到的是org/mortbay/servlets/CrossOriginFilter.class文件，把此文件复制到安装目录下的WEB-INF/CLASS文件夹中，在我电脑上的路径为：C:\Program Files (x86)\GeoServer 2.7.0\webapps\geoserver\WEB-INF\classes\org\mortbay\servlets\CrossOriginFilter.class.
2.修改geoserver配置文件
2.1 打开geoserver安装目录下的web.xml文件，我的电脑上路径为：C:\Program Files (x86)\GeoServer 2.7.0\webapps\geoserver\WEB-INF\web.xml
2.2 在filter集合末尾额外添加一个filter：
<filter>
<filter-name>cross-origin</filter-name>
<filter-class>org.mortbay.servlets.CrossOriginFilter</filter-class>
<init-param>
<param-name>allowedOrigins</param-name>
<param-value>*</param-value>
</init-param>
<init-param>
<param-name>allowedMethods</param-name>
<param-value>GET,POST</param-value>
</init-param>
<init-param>
<param-name>allowedHeaders</param-name>
<param-value>x-requested-with,content-type</param-value>
</init-param>
</filter>
<filter-name>cross-origin</filter-name>
<filter-class>org.mortbay.servlets.CrossOriginFilter</filter-class>
<init-param>
<param-name>allowedOrigins</param-name>
<param-value>*</param-value>
</init-param>
<init-param>
<param-name>allowedMethods</param-name>
<param-value>GET,POST</param-value>
</init-param>
<init-param>
<param-name>allowedHeaders</param-name>
<param-value>x-requested-with,content-type</param-value>
</init-param>
</filter>
2.3 在filter-mapping末尾额外添加一个filter-mapping
<filter-mapping>
<filter-name>cross-origin</filter-name>
<url-pattern>/*</url-pattern>
</filter-mapping>
<filter-name>cross-origin</filter-name>
<url-pattern>/*</url-pattern>
</filter-mapping>